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# stabilitymu

Stability factor μ of 2-port network

## Syntax

```[mu,muprime] = stabilitymu(s_params)
```

## Description

[mu,muprime] = stabilitymu(s_params) calculates and returns the stability factors μ and μ′ of a 2-port network. The input s_params is a complex 2-by-2-by-M array, representing M 2-port S-parameters.

[mu,muprime] = stabilitymu(hs) calculates and returns the stability factors for the network represented by the S-parameter object hs.

The stability factor, μ, defines the minimum distance between the center of the unit Smith chart and the unstable region in the load plane. The function assumes that port 2 is the load.

The stability factor, μ′, defines the minimum distance between the center of the unit Smith chart and the unstable region in the source plane. The function assumes that port 1 is the source.

Having μ > 1 or μ′  > 1 is the necessary and sufficient condition for the 2-port linear network to be unconditionally stable, as described by the S-parameters.

## Examples

Examine the stability of network data from a file:

```% Calculate stability factor and conditions
s_params = ckt.NetworkData.Data;
freq = ckt.NetworkData.Freq;
[mu muprime] = stabilitymu(s_params);
% Check stability criteria
stability_index = (mu>1)|(muprime>1);
is_stable = all(stability_index)
% List frequencies with unstable S-parameters
freq_unstable = freq(~stability_index);```

expand all

### Algorithms

stabilitymu calculates the stability factors using the equations

$\begin{array}{c}\mu =\frac{1-{|{S}_{11}|}^{2}}{|{S}_{22}-{S}_{11}^{*}\Delta |+|{S}_{21}{S}_{12}|}\\ \mu \prime =\frac{1-{|{S}_{22}|}^{2}}{|{S}_{11}-{S}_{22}^{*}\Delta |+|{S}_{21}{S}_{12}|}\end{array}$

where:

• S11, S12, S21, and S22 are S-parameters, from the input argument s_params.

• Δ is a vector whose members are the determinants of the M 2-port S-parameter matrices:

$\Delta ={S}_{11}{S}_{22}-{S}_{12}{S}_{21}$

• S* is the complex conjugate of the corresponding S-parameter.

The function performs these calculations element-wise for each of the M S-parameter matrices in s_params.

## References

Edwards, Marion Lee, and Jeffrey H. Sinsky, "A New Criterion for Linear 2-Port Stability Using a Single Geometrically Derived Parameter," IEEE Transactions on Microwave Theory and Techniques, Vol. 40, No. 12, pp. 2303-2311, December 1992.